The generalized principal eigenvalue for Hamilton–Jacobi–Bellman equations of ergodic type

Ichihara, Naoyuki

doi:10.1016/j.anihpc.2014.02.003

Cited by 22 publications

(36 citation statements)

References 22 publications

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“…Indeed, sinceū solves the first equation in (5.25) , we get−∆Φ + ∇H 0 (∇ū) · ∇Φ ≥ κ(−λ − κ|∇ū| 2 −m α )Φ.Using (5.27) andm → 0 as |x| → +∞, we obtain the claim. Reasoning as in[20, Proposition 4.3], or [28, Proposition 2.6], we get that R N e κūm dx < +∞, which concludes the estimate (5.26). With estimates(5.26) in force, the pointwise bounds stated in [28, Theorem 6.1] hold, namely there exist positive constants c 1 , c 2 , such that m(x) ≤ c 1 e −c2|x| on R N .…”

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confidence: 70%

Concentration of ground states in stationary mean-field games systems

Cesaroni

Cirant

2019

Analysis & PDE

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In this paper we provide the existence of classical solutions to stationary mean field game systems in the whole space R N , with coercive potential and aggregating local coupling, under general conditions on the Hamiltonian. The only structural assumption we make is on the growth at infinity of the coupling term in terms of the growth of the Hamiltonian. This result is obtained using a variational approach based on the analysis of the non-convex energy associated to the system. Finally, we show that in the vanishing viscosity limit mass concentrates around the flattest minima of the potential. We also describe the asymptotic shape of the rescaled solutions in the vanishing viscosity limit, in particular proving the existence of ground states, i.e. classical solutions to mean field game systems in the whole space without potential, and with aggregating coupling.

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confidence: 70%

Concentration of ground states in stationary mean-field games systems

Cesaroni

Cirant

2019

Analysis & PDE

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“…The quadratic case (a) has been proved in [7,Theorem 2.5], and the second claim in (b) (i.e., the case where 2 < m < ∞ and N ≥ 2) is also suggested by [8,Theorem 2.4] in a slightly different context. The essential novelty of this paper, compared with [7,8], lies in the simultaneous derivation of (b) and (c) in combination with the convergence result obtained in part A. In particular, claim (c) for N ≥ 2 can be derived by passing to the limit of (b) as m → ∞.…”

Section: Introductionmentioning

confidence: 89%

“…We refer, for instance, to [12] and references therein for more information on singular ergodic stochastic control and associated PDEs with gradient constraint. See also [6,7,8] for the stochastic control interpretation of λ m,β for 2 ≤ m < ∞. In this paper, we focus only on the PDE aspect and do not discuss its probabilistic counterpart.…”

Section: Introductionmentioning

confidence: 99%

Qualitative properties of generalized principal eigenvalues for superquadratic viscous Hamilton–Jacobi equations

Chasseigne

Ichihara

2016

Nonlinear Differ. Equ. Appl.

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This paper is concerned with the ergodic problem for superquadratic viscous Hamilton-Jacobi equations with exponent m > 2. We prove that the generalized principal eigenvalue of the equation converges to a constant as m → ∞, and that the limit coincides with the generalized principal eigenvalue of an ergodic problem with gradient constraint. We also investigate some qualitative properties of the generalized principal eigenvalue with respect to a perturbation of the potential function. It turns out that different situations take place according to m = 2, 2 < m < ∞, and the limiting case m = ∞.

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“…However, the critical value is not necessarily the optimal value. For more recent work on the relation of the critical value of an elliptic HJB equation of the ergodic type and the optimal value of the control problem see Ichihara (2015).…”

Section: The Hjb Equation For the Ergodic Control Problemmentioning

confidence: 99%

Controlled equilibrium selection in stochastically perturbed dynamics

Arapostathis¹,

Biswas²,

Borkar³

2018

Ann. Probab.

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We consider a dynamical system with finitely many equilibria and perturbed by small noise, in addition to being controlled by an 'expensive' control. The controlled process is optimal for an ergodic criterion with a running cost that consists of the sum of the control effort and a penalty function on the state space. We study the optimal stationary distribution of the controlled process as the variance of the noise becomes vanishingly small. It is shown that depending on the relative magnitudes of the noise variance and the 'running cost' for control, one can identify three regimes, in each of which the optimal control forces the invariant distribution of the process to concentrate near equilibria that can be characterized according to the regime. We also obtain moment bounds for the optimal stationary distribution. Moreover, we show that in the vicinity of the points of concentration the density of optimal stationary distribution approximates the density of a Gaussian, and we explicitly solve for its covariance matrix.MSC 2010 subject classifications: Primary 35R60; secondary 93E20.

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The generalized principal eigenvalue for Hamilton–Jacobi–Bellman equations of ergodic type

Cited by 22 publications

References 22 publications

Concentration of ground states in stationary mean-field games systems

Concentration of ground states in stationary mean-field games systems

Qualitative properties of generalized principal eigenvalues for superquadratic viscous Hamilton–Jacobi equations

Controlled equilibrium selection in stochastically perturbed dynamics

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